Quantum Metrology Triangle. François Piquemal

Transcription

1 Quantum Metrology Triangle François Piquemal

2 The Quantum Electrical Metrology group at the LNE SET&CCC: QHE&CCC: JE: Calculable Capacitor: Laurent Devoille, Nicolas Feltin, Sami Sassine Wilfrid Poirier, Félicien Schopfer, Laetitia Soukkassian Dominique Leprat Sophie Djordjevic, Olivier Séron, Hervé Ndilimabaka Pierre Gournay, Olivier Thévenot, Ludovic Lahousse For the quantum metrological triangle: - Collaboration CEA-Saclay & Grenoble, LPN/CNRS Marcoussis, PTB Braunschweig&Berlin REUNIAM project: METAS, MIKES, VSL, NPL, PTB Bonn, 4 November 2009

3 OUTLINE I) Introduction 1) Towards a quantum SI 2) Quantum electrical metrology II) Aims of the Quantum Metrological Triangle 1) Uncertainty thresholds 2) Determination of the charge quantum 3) SET standards III) QMT via Electron Counting Capacitance Standard: Q = CV IV) QMT Experimental set-up based on the CCC: U = RI 1) CCC: Cryogenic Current Comparator 2) Overall set-up at LNE and first results V) Conclusion

4 The present SI the website: bipm.org «The seven base units provide the reference used to define all the measurement units of the SI. As science advances, and methods of measurement are refine, their definitions have to be revised.» 1967/68: Definition of the second «Mise en pratique»: Cesium clock 1983: Definition of the meter by fixing the speed of light in vacuum «Mise en pratique»: Laser 1889: PtIr International Prototype of kilogram The only remaining artefact used to define a base unit in the SI! The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. It follows that the mass of the international prototype of the kilogram, m(k) is always 1 kg exactly Stability of the unit of mass?

5 The present SI 1948: Definition of the ampere by fixing the magnetic constant µ 0 The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to newton per meter of length. It follows that the magnetic constant µ 0, also known as the permeability of free space is 4π H/m exactly Biot-Savart Law : v B = μ 0 4π v I r ˆ dl B = μ I 0 r 2 2πd v F = I ( ) Lorentz Force Law : µ 0 = 4π 10-7 N/A 2 dl v B v F dl = IB = μ 0 I2 2πd (N/m) µ 0 /2π (A 2 /m) This definition of the ampere is difficult to be realised with enough accuracy. Today, its realization derives from the volt and the ohm ampere balance: the best accuracy was parts in 10 6

6 Towards a Quantum SI: a new SI well matched with the present knowledge of physics Fixing h and e and keep floating M(IPK) and µ 0 But, considering that the uncertainties from measurements were not at the required level, the Comité International des Poids et Mesures (CIPM ) called for Preparative steps towards new definitions of the kilogram, the ampere, the kelvin and the mole in terms of fundamental constants for possible adoption by the 24 th CGPM in 2011 and recommended that NMIs Constant Present SI CODATA 2006 ppb uncertainty 2011 quantum SI CODATA 2006 ppb uncertainty M (IPK) exact 50 h 50 exact N A 50 exact e 25 exact K J (2e/h) 25 exact R K (h/e 2 ) 0.68 exact F 25 exact m e m p u, m u 50 (amu) 1.4 (amu)... should pursue vigorously their work presently underway aimed at providing the best possible values of the fundamental constants needed for the redefinitions now being considered

7 Chain of SI realisations of electrical units and materialization Equivalence between electrical and mechanical powers Watt balance watt meter, kilogram, second µ 0 c 2 farad Thompson-Lampard theorem Calculable capacitor 10 pf ± 10 V Josephson effect volt ohm siemens Quantum Hall effect henry 100 Ω 1 ΜΩ coulomb ampere

8 Quantum standards: universality and high reproducibility (1/3) Unique representation of the volt and the ohm: The international comparisons of complete JE and QHE systems show a high level of consistency: from a few to a few Comparison of 10 V JAVS (r Lab-rBIPM)/r BIPM (10-9 ) BIPM.EM-K12 Bilateral comparison of QHRS with BIPM BNM-LCIE METAS PTB NPL RH R H (2) / 100 Ω 10k-100 kω / 100 Ω Ω / 1 Ω NIST 99-04

9 Quantum standards: universality and high reproducibility (2/3) Universality tests U J / f and i R H (i) independent of materials at a level of In microbridge planar Nb/Cu/Nb junction J.S. Tsai et al Si-MOSFET GaAs/AlGaAs A. Hartland et al 1991, B. Jeckelmann et al 1995 few 10-9 Graphene GaAs/AlGaAs A. Tzalenchuk et al I s L s V 1 V 2 I 1 I 2 L 1 L hf 2 These remarkable results from comparison and universality test do not prove that the phenomenological constants are exactly 2e/h and h/e 2 but they strengthen our confidence in the equalities K J = 2e/h and R K = h/e 2 in addition to strong theoretical arguments. If corrections exist, they will be probably of fundamental nature.

10 Quantum standards: universality and high reproducibility (3/3) Strong theoretical arguments supporting von Klitzing and Josephson relations: R K = h/e 2 and K J = 2e/h Today, from validation tests: R K = h/e 2 (1 + [24 ± 18] 10-9 ) K J = 2e/h (1 + [238 ± 720] 10 9 ) The large uncertainty on K J comes from the present discrepant values of Γ ph-90 (lo) and V m (Si) Calc. capacitor P. Mohr, B.N. Taylor, D.B. Newell, Rev of Mod Phys., vol.80, 2008

11 Chain of SI realisations of electrical units and materialization meter, kilogram, second µ 0 c 2 Single electron tunneling Watt balance watt farad Calculable capacitor 10 pf ± 10 V Josephson effect volt ohm siemens Quantum Hall effect henry 100 Ω 1 ΜΩ coulomb < 1 na Single electron tunneling ampere

12 The quantum metrological triangle (QMT) experiment By means of SET devices such as electron pumps, hybrid turnstile, a current standard with quantized amplitude is available : I = ef The experiment originally proposed by K. Likharev and A. Zorin in 1985 consists of applying Ohm s law, U = R I directly to the quantities issued from ac JE, QHE and SET. f U =n (h/2e) f I = e f SET pump I = e f R H V J J f J U Josephson effect Quantum Hall effect SET effect I I = (e 2 /h)u

13 Closing the triangle: first way U = R I As for JE, K J = (2e/h) JE and QHE, R K = (h/e 2 ) QHE, one can define a phenomenological constant: Q X = e SET Dimensionless product to be measured JE U =n K J -1 f J R K K J Q X = 2 if : Exactness?! R K = h/e 2 K J = 2e/h Q X = e U R K K J Q X = (ni/g) ( f J / f SET ) DC R I G = N P /N S gain of the CCC QHE R = R K /i SET I = Q X f SET

14 The QMT experiment Another approach to close the triangle is to apply Q = CV Charging a capacitor electron per electron with a pump measuring the voltage drop with Josephson voltage standard, calibrating the capacitance by means of QHR standard Q = Ne SET pump controlled by an electrometer C V J J f J Electron counting capacitance standard (ECCS)

15 Closing the triangle: second way Q = C U ECCS: Charging a capacitor electron per electron by a SET pump and measuring the voltage drop with Josephson standard C ECCS =K J Q X /[(n/n) f J ] C ECCS compared to C X calibrated by means of QHR standard via quadrature bridge (RCω = 1) JE U =n K J -1 f J U R K K J Q X = (n/n)(c ECCS /C X ) ( f J / f q ) i Quadrature bridge AC C Q QHE C = i/(2πf q R K ) SET N Q X

16 OUTLINE I) Introduction 1) Towards a quantum SI 2) Quantum electrical metrology II) Aims of the Quantum Metrological Triangle 1) Uncertainty thresholds 2) Determination of the charge quantum 3) SET standards III) QMT via Electron Counting Capacitance Standard: Q = CV IV) QMT Experimental set-up based on the CCC: U = RI 1) CCC: Cryogenic Current Comparator 2) Overall set-up at LNE and first results V) Conclusion

17 Aims of the QMT (1/3) To confirm with a very high accuracy that these three effects of condensed matter physics give the free space values of constants 2e/h, h/e 2 and e. The ultimate target uncertainty is one part in 10 8 If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e 2 and e will be strengthened. If deviation occurs, some other works both experimental and theoretical will have to be done.

18 Different uncertainty thresholds for closing the QMT First critical test of validity for SET: Uncertainty of 1 ppm Neither the JE nor the QHE is questionable at that uncertainty level recently completed by NIST with σ = 9.2 parts in 10 7 Second uncertainty level lies between 7 parts in 10 7 and 2 parts in 10 8 Resulting information will be mainly relevant for the JE and the SET

19 Aims of the QMT (2/3) To confirm with a very high accuracy that these three effects of condensed matter physics give the free space values of constants 2e/h, h/e 2 and e. The ultimate target uncertainty is one part in 10 8 If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e 2 and e will be strengthened. If deviation occurs, some other works both experimental and theoretical will have to be done. To determine the elementary charge e or, in other words, the charge quantum brought by the SET devices Indeed, one can show that by combining the three experiments QMT, calculable capacitor and watt balance, a determination of e involved in SET devices without assuming that R K = h/e 2 and K J = 2e/h is possible.

20 Determination of the charge quantum (1/3) 1897: J.J. Thomson Discovery of the electron : R.A. Millikan Charged oil droplets method e/e ref direct values ajdusted values e ref = C u r F = N A e 1928 : E. Bäcklin The value e is derived from known value of Faraday constant and the determination of N A by x-ray method > 1940, e is derived from a complex calculation and is no more related to an experiment. 1929: Raymond T. Birge Least-Squares Adjustments of the Constants - Concept of most probable value; - Interconnection of constants by well established relation; - A precise knowledge of only a limited number of constants and relations was needed to evaluate the values of a large group with sufficient accuracy - The willing of labs to share their works 1952: Dumond and Cohen Increasing number of data and observational equation 1969: CODATA Task group on fund. constants was formed 1973: 1 st LSA from CODATA

21 Determination of the charge quantum (2/3) In the framework of the LSA by the CODATA (>1973), e is no more an adjustable constant and its value is obtained from the relation giving α: µ 0 c α = e = 2αh 2h/e 2 µ 0 c CODATA 2006: e = C and σ = α from a e, h/m and R K (Calc. capacitor + QHE) h via K J2 R K (WB + QHE + JE) In fact, values of e can be derived directly from experiment, but with a larger uncertainty

22 Determination of the charge quantum (3/3) Two direct values independent of the QHE and the JE (e - e CODATA 2006 ) x 10 6 Direct value LSA&CODATA e direct e = [α 3 A r (e)m u /(µ 0 R N A )] 1/2 - A r (e): 2006 CODATA - M u : = 10 3 kg.mol 1 exactly. - R : 2006 CODATA - α : a e and h/m Cs, Rb - N A N A = V m (Si)/(8 1/2 d 2203 ) e CODATA 2006 = (40) C Comparisons: Q X e, R K h/e 2, K J 2e/h Q X Q X value from QMT at NIST with ECCS σ =

23 Aims of the QMT (3/3) To confirm with a very high accuracy that these three effects of condensed matter physics give the free space values of constants 2e/h, h/e 2 and e. The ultimate target uncertainty is one part in 10 8 If there is no deviation, our confidence on the three phenomena to provide us with 2e/h, h/e 2 and e will be strengthened. If deviation occurs, some other works both experimental and theoretical will have to be done. To determine the elementary charge e or, in other words, the charge quantum brought by the SET devices Last but not least, to establish whether the SET can achieve a high level quantization and lead to a natural mise en pratique of the future definition of the ampere New quantum electrical standards

24 SET standards New definition of the ampere based on fixed value of e: The ampere is the electrical current scaled such that the elementary charge is exactly C An alternative (but explicit unit definition instead of explicit constant definition): The ampere is the electrical current equivalent to the flow of exactly 1/( ) elementary charges per second. It follows that this definition fixes the elementary charge as exactly A s New quantum electrical standards: - Quantum current standard for the calibration domain < 1 na Direct traceability for electrical current instead of the present route from JAVS&QHRS Q = Ne - Natural capacitance standard Quantum standard like JAVS & QHR C = Q / V (2e 2 /h) f electron pump C V

25 OUTLINE I) Introduction 1) Towards a quantum SI 2) Quantum electrical metrology II) Aims of the Quantum Metrological Triangle 1) Uncertainty thresholds 2) Determination of the charge quantum 3) SET standards III) QMT via Electron Counting Capacitance Standard: Q = CV IV) QMT Experimental set-up based on the CCC: U = RI 1) CCC: Cryogenic Current Comparator 2) Overall set-up at LNE and first results V) Conclusion

26 Cryogenic Capacitor C cryo 1.8 pf N2 Principle of ECCS 1) Pumping electrons on and off cryogenic capacitor N1 V FB island 1 ff ΔV V FB (V) V e E. Williams et al, Journal of NIST, 1992, M. Keller et al, Science Time (s) C J 0.2 ff Each ΔV gives a value of C cryo = Ne/ΔV 2) Comparing capacitances V G1 V G2 electrometer 1.6 khz C cryo = 1.8 pf V G5 7-J pump 10 pf stray capacitance V 1 ND N2 N1 island T < 100 mk V G6 V 2 C ref = 10 pf

27 Present status of Q = CV triangle experiments 1) NIST First comparison of the C cryo values by counting electrons and by a capacitance bridge M. Keller et al, Science, vol 285, 1999 Q X /e 1 = (-0.09 ± 0.92) 10-6 M. Keller et al, Metrologia ppm 2) PTB talk of M. Wulf poster of B. Camarota Progress on electron counting capacitance standard

28 OUTLINE I) Introduction 1) Towards a quantum SI 2) Quantum electrical metrology II) Aims of the Quantum Metrological Triangle 1) Uncertainty thresholds 2) Determination of the charge quantum 3) SET standards III) QMT via Electron Counting Capacitance Standard: Q = CV IV) QMT Experimental set-up based on the CCC: U = RI 1) CCC: Cryogenic Current Comparator 2) Overall set-up at LNE and first results V) Conclusion

29 Principle of the CCC Harvey 1972 δφ few µφ 0 /Hz 1/2 (typ.) I 1 I 2 I B=0 (a) Application of the Ampère s law : (a) B.dl = 0 = µ 0 (I 1 +I 2 -I) I = I 1 +I 2 SQUID X X Superconducting shield Pick-up coil Φ = f(i) I I ampere - turn balance : I = 0 N 2 I 2 = N 1 I 1 G = I 2 /I 1 = N 1 /N 2 I 2 I 1 Β = f(i) Supercurrent Secondary I = N 1 I 1 -N 2 I 2 Primary winding winding N N 1 I 2 I 2 1

30 «The snake swallowing its tail!» Hergé, Tintin au Congo Les éditions du petit 20 ème 1931 Castermann

31 Experimental set-ups based on CCC CCC as current amplifier Two detection levels: magnetic flux and voltage Highly accurate gain: σ G < 10-9 SET pump I SET N 1 δφ N 2 GI SET R H V δv J δv/v = ([δi 2 CCC+ (4kT/G 2 R H )]/I 2 SET+ δv 2 ND/V 2 ) 1/2 δi CCC /I SET CCC (G = ): δi < 1 fa/hz 1/2, t meas = 10 hours, σ <I> /I < 10-7 I > 80 pa CCC as current detector Single detection level δi/i = [4kT/R cryo )] 1/2 /I SET R cryo = 100 MΩ at 4.2 K: σ <I> /I < 10-7 I > 80 pa SET pump δι I SET I R N 1 δφ N 2 J R cryo Output in both cases: SQUID operates at δφ 0 (Flux Locked Loop)

32 Present CCCs with DC SQUID CCCs in use Φ ext = 49 mm, Φ int = 22 mm, h = 13 mm G max = S CCC th = 4.5 µa t/φ 0, δi th = 0.7 fa/hz 1/2 - S CCC meas = 5 µa t/φ 0, δi meas = 4 fa/hz 1/2 δi meas = 5 fa/hz 1/2 at 1 Hz connected to R pump - Winding ratio error : ε dc = , ε ac = Self resonances Spectral density of noise, config. 3 f res = 3.5 khz ( turns) 7.4 khz 3.5 khz

33 Principle of the QMT set-up at LNE 10 MHz rubidium reference Waveform generator DC SQUID 70 GHz source π/2 δφ Flux Transformer INT FB R FB V out EXT FB 30 mk SET pump N 1 turns CCC N 2 turns Calibrated 10 kω V R V d 4.2 K 4.2 K Program. INT FB: non accurate mode JAVS EXT FB: accurate mode V J DAC based Bias current source

34 Results on 3-junctions R pump (SQUID in int FB) 3 junctions R pump fabricated by the PTB A. Zorin et al., JAP2000 S. Lotkhov et al, APL, vol.78, 2001 source R Cr (25-50 kω) drain V g1 V g2 gates Measurement of current steps up to 100 MHz! B. Steck et al, Metrologia 2008 I (pa) 40 fa

35 ΔI /I Spring 2009: precise measurement of flatness (int FB mode) See poster Sami Sassine et al Step +, 10 MHz V p (µv) ΔI /I V P + V p (µv) fa ΔI /I Step V p (µv) ΔI /I V p (µv) fa The plateaux are flat over 60 µv (σ mean = 2 parts in 10 5 ) Better and reproducible white noise level 5 fa/hz 1/2

36 Measurements with SQUID in ext FB 1.7 u r = Q X = (N 2 /N 1 )(n J f J /K J V d )/(f SET R) Δe/e = (Q X - e CODATA )/e CODATA Δe/e (*10-4 ) Weighted mean 5.E-04 4.E-04 3.E-04 f SET =23.55 MHz n J =5 u r =7x10-6 f SET =22.65 MHz n J =5 1.2 f SET = MHz n J = Number of measurement B. Steck et al. CPEM 08 (Qx-e)/e 2.E E E-04 1.E-04 0.E+00 Weighted average u r =1.3x E Number of measurement Δe /e 1.2E E E E E E-05 f SET =37.69 MHz n J =8-1.2E # 1.5x10-4 Discrepancies between 2 runs of measurements

37 V- Conclusion: contribution of the QMT to test and hopefully to enhance our confidence on QHE, JE and SET to provide h/e 2, 2e/h and e to improve knowledge on fundamental constants, in particular the elementary charge The direct determination of e via QMT, calculable capacitor and watt balance links up with the historical experiment of Millikan, early last century to give some elements of thought in the frame of the discussion around the reform of the SI to advocate the mise en pratique of the definition of some electrical units with SET standards

38 Outlook of the QMT To complete the development of new CCCs with the aim at reaching : δi < 1 fa/hz 1/2 To investigate single charge transport devices with I >> 1 pa, for example: - Hybrid SINIS SET turnstile (MIKES/TKK) - Pump based on Si or GaAs/AlGaAs nanowire (CEA, PTB) REUNIAM! Current plateaux observed at a level 100 pa - Parallelization of devices to increase much more the current To close the triangle - in short term, at a level of 1 part in 10 6, - in mid term, at a level of parts in 10 7, - in long term down to part in 10 8 and less depending on the SCT current sources!

39 Thank you very much!

40 Evolution of national prototypes over 120 years ΔM (µg) / /92 Possible drift of IPK: per century!

41 Cross calculable capacitance standard Theorem of A. Thompson and D. Lampard (1956): For a cylindrical system of 4 isolated electrodes of infinite length and placed in vacuum, exp(-π γ 13 /ε 0 ) + exp(-π γ 24 /ε 0 ) = 1 1 γ 13 γ 24 2 In the case of a perfect symmetry with identical γ ij 4 3 γ 13 = γ 24 = γ = (ε 0 ln2)/π = pf/m ΔC = γδl LNE, five-electrodes capacitor 2000 From the farad to the ohm METRE Calculable capacitor QHE QHE R K 2012 a) d) f) e) b) c) 1 pf 10 pf Capacitance bridges 100 pf 1000 pf 10 nf SECOND R C ω = 1 ω (rd/s) R 100 Ω Calculable CCC resistor DC D C Quadrature bridge AC 10 kω

42 Josephson voltage standards Brian Josephson 1962 Quantum effects occurring between two superconducting electrodes separated by a small region where the superconductivity is weakened: thin insulating film Microwave radiation f I n = 1 10 V Josephson junction arrays Oxide layer 2 nm V I (ma) V = = (h/2e) f/k f J 145 µv 145 µv at f = 70 GHz at f= 70 GHz K J = 2e/h, the Josephson constant Programmable arrays for DC and AC applications See Review: B. Jeanneret, S. Benz, EPJST 172, 2009 Steps around 10 V

43 Quantum Hall resistance standards Klaus von Klitzing, 1980 At low temperature and under high magnetic field, the Hall resistance of the 2DEG exhibits plateaux centred on quantized values: R H (i) = h/ie 2 = R K /i, where i is an integer, R K the von Klitzing constant. LEP 514 Hall bar sample GaAs/AlGaAs heterostructure 0 Arrays of Hall bars for scaling up to 1.29 MΩ and down to 100 Ω Review: W. Poirier, F. Schopfer EPJST 172, 2009 Quantum Wheatstone bridge for ultra high precise comparison (10-11 ) Quantum impedance standard: on 10 pf J. Schurr et al. Metrologia 2009 R H C3 (B +) (Ω) T=1,3K I=100 µa QHARS 129 QHARS B(T) R H1 R H4 I ub. V R H2 R H R xx a (Ω)

44 Watt balance: monitoring the kilogram electrically B F z I Static P m v B Dynamic ε speed v LNE balance F z = mg = -I Φ/ z ε = - Φ/ t = - Φ/ z v Equivalence between mechanical and electrical power F z v = εi mgv = εv/r - ε and V in terms of Josephson effect: ε = n 1 f 1 /K J, V = n 2 f 2 /K J -Rin terms of quantum Hall effect: R = R K /i 4mgv h =, assuming K J = 2e/h and R K = h/e A 2 m(ipk) = h A 4gv A mgv = K 2 JR K where A = n 1 f 1 n 2 f 2 i Monitoring the kilogram in term of h

45 Single electron tunneling: towards quantum current standard Electron box U C J + - Tunnel junction n Metallic island q = Q G - Q = ne Coulomb Blockade of the tunneling events when n -1/2 < C G U/e < n + 1/2 Average box charge [e] <n> 4.0 Experiment Theory Coulomb staircase C g V g [e] C G U/e C G Gate capacitor - Thermal fluctuations of n are negligible if k B T << e 2 /C i - Quantum fluctuations of n negligible when R j >> R K The wave function of electron in excess on the island is well localised 3-junctions electron pump Two modulation signals transistor at frequency f phase-shifted by Φ =π/2 I +V/2 n 1 U 1 C 1 n 2 U 2 C 2 -V/2 I = e f For f = 100 MHz I = 16 pa AnimPompe.exe e (0,1) N P e (0,0) (1,0) Minimum energy states of of the the island pump As as a function of of U 1 and 1 and U 2 2 Unwanted effects and error corrections talk of M. Wulf -e -e C 2 U 2 C 1 U 1 (n 1,n 2 )

46 Observational equations taken into account in the adjustment of values of fundamental constants by CODATA A = f(x,y,..) measured quantity adjusted constants

47 Toward an improved calibration of small currents SET current standard Sub-nano ammeters (integration bridge, electronic device) Metrology (elect., ionising radiat.): Calibration of - low-current standard source - high-value resistance standard Electronic industry: - Characterization of components relative standard uncertainty (1σ) 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 integration bridge SET current standard potential difference 1E-18 1E-15 1E-12 1E-09 1E I (A) Existing methods RT comparator QHE & JV standards Physical size electrical signals such as current